On Frobenius Theorem and Classication of 2-Dimensional Real Division Algebras

Home , Algebra over a field, Division algebra

U.U.D.M. Project Report 2020:18

Mikolaj Cuszynski-Kruk

Examensarbete i matematik, 15 hp Handledare: Martin Herschend Examinator: Veronica Crispin Quinonez Juni 2020

Department of Mathematics Uppsala University

Abstract

A proof of Frobenius theorem which states that the only ﬁnite-dimensional real associative division algebras up to isomorphism are R, C and H, is given. A complete list of all 2-dimensional real division algebras based on the multiplication table of the basis is given, based on the work of Althoen and Kugler. The list is irredundant in all cases except the algebras that have exactly 3 idempotent elements. Contents

1 Introduction 1

2 Background 2 2.1 Rings ...... 2 2.2 Linear algebra ...... 4 2.3 Algebra ...... 7

3 Frobenius Theorem 10

4 Classiﬁcation of ﬁnite-dimensional real division algebras 12 4.1 2-dimensional division algebras ...... 13

References 22 1 Introduction

An algebra A over a field F is vector space over the same field together with a bilinear multiplication. Moreover, an algebra is a division algebra if for all non-zero a in A the maps La : A → A defined by v 7→ av and Ra : A → A defined by v 7→ va are invertible. If an algebra has a unit and the multiplication is associative then the algebra has a ring structure. In this paper we will only study algebras over the field of real numbers which are called real algebras, moreover we shall assume that the vector space is finite-dimensional. A trivial example of a real division algebra is the field of real numbers with standard multiplication. The study of real division algebras originate form the study of number sys- tems. The construction of the field of complex numbers yielded a 2-dimensional real division algebra. Further developments were made by Sir William Rowan Hamilton. After studying the complex numbers he tried to construction a 3- dimensional real division algebra and according to a famous story, after many failed attempts, during a walk Hamilton came up with a 4-dimensional construction instead, defined by the property

i2 = j2 = k2 = ijk = −1 which later would be called the quaternion algebra. Although Hamilton suc- ceeded in constructing a 4-dimensional division algebra there was a price to be paid, the quaternions are not commutative. For further history of Hamilton and the quaternions see [2]. In the same way complex numbers can be seen as pairs of real numbers with multiplication deﬁned by (1), the quaternions can be see as pairs of complex numbers with multiplication also deﬁned by (1).

(z1, z2) · (w1, w2) = (z1w1 − w2z2, z2w1 + w2z1). (1)

By continuing this line of thoughts, pairs of quaternions are a perfect candidate for a 8-dimensional real division algebra. Indeed, they form the so called octonions discovered independently by Graves [8] and Cayley [5]. Similarly to the quaternions not being commutative the octonions are in addition not associative. One could think that it is possible to construct 2n−dimensional real division algebras by continuing this process, but it is not the case. In 1958 Bott and Milnor [4] proved that the only possible dimensions of a real division algebra are 1,2,4 and 8. In 1878 Frobenius proved that the only real associative division algebras up to isomorphism are R, C and H, a proof will be given in Section 3. Later in 1931, Zorn [12] proved that by weakening the assumption of associativity to instead only require that the algebra is alternative yields only one additional algebra, the octonion algebra. When considering arbitrary real division algebras, the classiﬁcation of 2-dimension real division algebras is known and will be given in Section 4.1. For

1 the 4- and 8-dimensional case only some special cases have been classiﬁed. An overview of some classiﬁcations is given in [6].

2 Background

In this section we deﬁne useful notions and establish results that will be necessary in order to prove Frobenius theorem.

2.1 Rings Basic theory of rings will be needed in order to understand properties of an associative algebra. The following subsection is based on [11, Chapter 2]. A natural way to start is to deﬁne the notion of a ring. Deﬁnition 2.1. A ring is a 5-tuple (R, +, ·, 0, 1) consisting of a set R, two binary operation + and · that are closed in R and two elements 0, 1 ∈ R such that for all a, b, c ∈ R

(a + b) + c = a + (b + c), (A1) a + b = b + a, (A2) a + 0 = 0 = 0 + a, (A3) ∃ −a ∈ R a + (−a) = 0 = −a + a, (A4) (a · b) · c = a · (b · c), (M1) a · 1 = a = 1 · a, (M2) a · (b + c) = a · b + a · c, (D1) (b + c) · a = b · a + c · a. (D2)

Remark 2.1.1. (R, +, ·, 0, 1) will often be denoted simply by R and a · b by ab. Example 2.1.2. (C, +, ·, 0, 1) where the operations are interpreted as standard addition and multiplication of complex numbers, is a ring.

Deﬁnition 2.2. (S, +, ·, 0, 1) is called a subring of a ring (R, +, ·, 0, 1) if S ⊆ R and the following holds for all a, b ∈ S

0, 1 ∈ S, (S1) a + b ∈ S, (S2) −a ∈ S, (S3) ab ∈ S. (S4)

Remark 2.2.1. In order to see if S satisﬁes 0 ∈ S, (S2) and (S3) it is enough to check if S is non-empty and that for all x, y in S, x − y is also in S. Since then x in S implies that 0 = x − x is in S and for all x, y in S, −x = 0 − x is also in S and hence so is y + x = y − (−x).

2 Since Frobenius theorem concerns division algebras the ability to perform division is important. Division is deﬁned as the inverse of multiplication but an ring does not required the existence of multiplicative inverses and rings that does have multiplicative inverses are called division rings. Deﬁnition 2.3. A division ring is a ring in which every non-zero element has an inverse, i.e. a ring R such that

∀x ∈ R \{0} ∃y ∈ R xy = 1 = yx.

Proposition 2.4. For each x ∈ R it’s inverse is unique and denoted x−1.

Proof. Assume y1, y2 are inverses of x, then y1x = 1 = xy2 and y1 = y1(xy2) = (y1x)y2 = y2. The quaternions were ﬁrst constructed by Hamilton and can be seen as an extension of complex number by two additional imaginary units, j and k, that satisfy i2 = j2 = k2 = −1 = ijk. Hamilton’s construction was of a geometrical nature but there is another way of constructing quaternions, namely as a subset of M2×2(C), i.e the set of 2 × 2 matrices with complex entries. z w Proposition 2.5. The set = : z, w ∈ is a subring of M ( ). H −w z C 2×2 C

Proof. For all z, w ∈ C it holds that z − w = z − w hence if Z,W ∈ H then 1 0 Z − W ∈ . The multiplicative identity in M ( ) is I = and since H 2×2 C 0 1 x y z w 1 = 1 and 0 = 0, I is in . Moreover , ∈ implies H −y x −w z H

x y z w xz − yw xw + yz xz − yw xw + yz = = −y x −w z −yz − xw −yw + xz −xw + yz xz − yw since z w = zw for all complex numbers. Hence H is closed under multiplication and thus a subring of M2×2(C).

Remark 2.5.1. Note that every matrix in H is of the form √ √ x1 + x2 −1 x3 + x4 −1 H 3 √ √ −x3 + x4 −1 x1 − x2 −1 √ √ 1 0 −1 0 0 1 0 −1 = x + x √ + x + x √ 1 0 1 2 0 − −1 3 −1 0 4 −1 0

1 0 where x , x , x , x ∈ , moreover if we denote the matrices , 1 2 3 4 R 0 1 √ √ −1 0 0 1 0 −1 √ , and √ by 1, i, j and k, respectively, 0 − −1 −1 0 −1 0 then it can easily be checked that i2 = j2 = k2 = −1 = ijk holds.

3 Proposition 2.6. The ring of quaternions is a division ring. z w Proof. Let ∈ \{0} then −w z H

1 z −w 1 z −w = ∈ H and |z|2 + |w|2 w z |z|2 + |w|2 −−w z 1 z w z −w 1 0 = . |z|2 + |w|2 −w z w z 0 1

Remark 2.6.1. Note that the equation i2 = j2 = k2 = −1 = ijk determine the multiplication in H completely. Deﬁnition 2.7. The centre of a ring R is the subset

Z(R) = {a ∈ R : ∀b ∈ R, ab = ba}.

Lemma 2.8. If r ∈ R then r1 ∈ Z(H). r 0 Proof. Let r ∈ then r1 = and for all z, w ∈ , R 0 r C

r 0 z w rz rw z w r 0 = = . 0 r −w z −rw rz −w z 0 r

The notion of a fields will be useful in the next subsection and the following lemma is a technical one. Definition 2.9. A field is a non-zero division ring in which the multiplication commutes, i.e. a division ring R 6= {0} such that ∀x, y ∈ R, xy = yx.

Lemma 2.10. Let F be a ﬁeld. If xy = 0 for some x, y ∈ F then then x = 0 or y = 0. Proof. If x is non-zero and xy = 0, then since F is a ﬁeld x has an inverse x−1.

0 = xy ⇔ 0 = x−1 · 0 = x−1xy = y.

Hence x = 0 or y = 0 must hold.

2.2 Linear algebra Vector spaces are another important part of algebras hence an overview of some basis properties is in order. This subsection is based on [3, Chapters 1, 2 & 3.A].

4 Definition 2.11. A set V together with two binary operations is called a vector space over a field F if the two operations + : V × V → V and · : F × V → V satisfies the following axioms for all u, v, w ∈ V and λ, µ ∈ F

u + (v + w) = (u + v) + w, (A1) u + v = v + u, (A2) ∃0 ∈ V, 0 + v = v = v + 0, (A3) ∃ −v ∈ V, v + (−v) = 0 = −v + v, (A4)

1F · v = v, (M1) λ(µv) = (λµ)v, (M2) λ(u + v) = λu + λv, (D1) (λ + µ)v = λv + µv. (D2)

Remark 2.11.1. In the rest of this section all vector spaces will be over a field F if not stated otherwise. Definition 2.12. A subset W of a vector space V is called a subspace if W satisfies

0 ∈ W, (S1) ∀u, v ∈ W, u + v ∈ W, (S2) ∀v ∈ W, λ ∈ F, λv ∈ W. (S3)

Remark 2.12.1. A subspace is a vector space and all vector spaces have {0} as a subspace. In order to be able to easily describe elements of a vector spaces the following notions are important. Deﬁnition 2.13. Let V be a vector space. The span of a set of vectors {vi ∈ V : i = 1, . . . , n} is the set

span{v1, . . . , vn} = {λ1v1 + ... + λnvn : λ1, . . . , λn ∈ F }.

If moreover span{v1, . . . , vn} = V then v1, . . . , vn spans V . If the set consist of only one vector v then the span can also be denoted by vF . Remark 2.13.1. The span is a subspace.

Deﬁnition 2.14. A collection of vectors {v1, . . . , vn} is called linearly independent if the equation

λ1v1 + ··· + λnvn = 0, λi ∈ F has only the trivial solution, λ1 = ··· = λn = 0. Otherwise the vectors are called linearly dependent. Deﬁnition 2.15. A set of vectors B is called a basis of a vector space V if B is a linearly independent set and B spans V .

5 Definition 2.16. If a vector space V has a basis B then the dimension of V is defined as the cardinality of B and denoted dim V = |B|. Remark 2.16.1. The dimension is well-defined since if a vector space V has a basis then the cardinalities of all bases of V are equal, for proof see [3, Theorem 2.35]. The notion of direct sum will be useful when constructing division algebras and in order to define a vector space structure on the quaternions.

Deﬁnition 2.17. Let U1,...,Un be subspaces of a vector space V . The sum

U1 + ··· + Un = {u1 + ··· + un : ui ∈ Ui, i = 1, . . . , n} is called a direct sum of U1,...,Un if each element can be written as a sum of ui:s in exactly one way and is denoted U1 ⊕ · · · ⊕ Un.

Lemma 2.18. Let V be a vector space with basis B = {v1, . . . , vn} then the sum v1F + ··· + vnF is a direct sum and v1F ⊕ · · · ⊕ vnF = V .

Proof. Let u, w ∈ v1F + ··· + vnF and I = {1, 2, . . . , n} then there exists λi, µi ∈ F for all i ∈ I such that u = λ1v1 +···+λnvn and w = µ1v1 +···+µnvn. Assume that u = w. Then

0 = u − w = (λ1 − µ1)v1 + ··· + (λn − µn)vn and since v1, . . . , vn are linearly independent λi = µi must hold for all i ∈ I. Thus every element has a unique representation and the sum is a direct sum. For the equality let v ∈ V , then since B spans V there exists λi ∈ F for all i ∈ I such that v = λ1v1 + ··· + λnvn and so v ∈ v1F ⊕ · · · ⊕ vnF . On the other hand viF is a subspace of V and hence a subset, so v ∈ viF implies that v ∈ V and the equality follows from the fact that V is closed under addition.

Example 2.18.1. C is a vector space over R with basis {1, i} so C = R ⊕ iR. Corollary 2.19. The ring of quaternions has a real vector space structure with basis {1, i, j, k}.

Proof. From Remark 2.5.1 it is clear that the set {1, i, j, k} spans H and is linearly independent hence it is a basis and thus H = R ⊕ iR ⊕ jR ⊕ kR. Definition 2.20. A map L from a vector space V to a vector space W , both over a field F , is called a linear map is it satisfies

∀u, v ∈ VL(u + v) = L(u) + L(v), (L1) ∀v ∈ V, ∀λ ∈ FL(λv) = λL(v). (L2)

Deﬁnition 2.21. Two vector space V and W are called isomorphic if there exists a bijective linear map L : V → W .

6 The rest of this section is based on [11, Sections 6.1-4] and contains useful tools for finding basis of a vector space. Definition 2.22. A bilinear map from a vector space V to a vector space W is a map B : V ×V → W such that for all v ∈ V the maps Rv : V → W, u 7→ B(u, v) are linear and for all u ∈ V the maps Lu : V → W, v 7→ B(u, v) are linear. If moreover B(u, v) = B(v, u) holds for all u, v ∈ V then B is called symmetric and if W is the underlying field of V then B is called a bilinear form. Definition 2.23. A symmetric bilinear form B on a real vector space V is called positive-definite if B(v, v) ≥ 0 for all v ∈ V with equality only when v = 0. Definition 2.24. A quadratic form on a vector space V is a map Q : V → F such that, ∀v ∈ V, ∀λ ∈ F,Q(λv) = λ2Q(v) and (Q1) B(u, v) = Q(u + v) − Q(u) − Q(v) is a bilinear form. (Q2) Definition 2.25. Let V be a real vector space. Two vectors u, v ∈ V are called orthogonal with respect to a positive-definite symmetric bilinear form B if B(u, v) = 0. Moreover, a set of vectors S is called orthogonal if all vectors in S are pairwise orthogonal.

Lemma 2.26. Let v1, . . . , vn be orthogonal non-zero vectors then they are linearly independent.

Proof. Let v1, . . . , vn be non-zero and orthogonal vectors with respect to a positive-deﬁnite symmetric bilinear form B. If λ1v1 + ··· + λnvn = 0 then,

0 = B(0, vi) = B(λ1v1 + ··· + λnvn, vi) = λiB(vi, vi) for all i = 1, . . . , n. The positive-deﬁniteness of B yields that λ1 = ··· = λn = 0.

2.3 Algebra In this subsection we define the notion of an algebra and prove that the quaternions are a real division algebra. But first some technical results that are taken from [9, Section III.8]. Definition 2.27. Let F be a field. A non-zero polynomial P ∈ F [X] of degree at least 1 is called irreducible if whenever P (X) = Q(X)R(X) for some Q, R ∈ F [X] then either Q or R is constant. Theorem 2.28 (The fundamental theorem of algebra). Every complex polynomial P of degree n ≥ 1 has a factorisation

P (X) = c(X − r1)(X − r2) ... (X − rn) where c, r1, . . . , rn ∈ C. The factorisation is unique up to permutation of the factors.

7 We will not prove this theorem, for a proof see [7, pages 4 & 188]. Corollary 2.29. A real polynomial is irreducible if and only if it has degree 1 or degree 2 and no real roots.

Proof. If a polynomial P of degree 1 is equal to Q(X)R(X) for some Q, R ∈ R[X] then 1 = deg P = deg Q + deg R hence either Q or R has degree 0, meaning that it is constants, so P is irreducible. If P has degree 2 and no real root then if P is a product of two non-constant polynomials then both must have degree r1 1 hence, P (X) = (c1X − r1)(c2X − r2) for some c1, c2, r1, r2 ∈ but then R c1 is real and a root of P , a contradiction, hence P is irreducible. Conversely, assume that P is irreducible and has degree n then by Theorem 2.28, P (X) = c(x − r1) ... (x − rn) where r1, . . . , rn are roots of P , moreover since P has real coeﬃcients P (X) = P (X) = c(X − r1) ... (X − rn). Now the uniqueness of factorisation yields that the sets of roots are equal, {r1, . . . , rn} = {r1,..., rn}, hence for all i = 1, . . . , n either ri is real or ri = rj for some j, in which case

2 (X − ri)(X − ri) = X − (ri + ri) + riri ∈ R[X] 2 since ri + ri = 2Re(ri), riri = |ri| ∈ R. Let 2m be the number of non-real roots. Then P is a product of m real polynomials of degree 2 and n − 2m real polynomials of degree 1, but since P is irreducible one of the following two cases must hold, m = 1, n − 2m = 0 or m = 0, n − 2m = 1. The first case implies that n = 2 and P has no real roots and the second implies that n = 1. Remark 2.29.1. Note that if X2 − 2pX + q is a real polynomial then it’s roots are X = p ± pp2 − q meaning that a monic degree 2 polynomial has no real roots and hence is irreducible if and only if p2 < q. Finally we are ready to define the notion of an algebra. The rest of this section is based on [11, Sections 7.1 & 7.7]. Definition 2.30. An algebra A over a field F is a vector space together with an operation · :(u, v) 7→ uv that is bilinear, i.e. for all u, v, w ∈ A, λ ∈ F, (u + v)w = uw + vw, (A1) u(v + w) = uv + uw, (A2) λ(uv) = (λu)v = u(λv). (A3) If moreover A together with + and · is a ring then A is called an associative algebra. Definition 2.31. If A and B are two algebras both over F then a map ϕ : A → B is called an algebra-homomorphism if for all u, v ∈ A and λ ∈ F , ϕ(λv) = λϕ(v), (I1) ϕ(u + v) = ϕ(u) + ϕ(v), (I2) ϕ(uv) = ϕ(u)ϕ(v). (I3) If moreover the map is bijective then ϕ is called an algebra-isomorphism.

8 Remark 2.31.1. If A and B are associative algebras then ϕ(1A) = 1B and ϕ is a ring-homomorphism. An algebra-isomorphism will often be called an isomorphism. Definition 2.32. The dimension of an algebra is defined as the dimension of the underlying vector space, if moreover the dimension is finite then the algebra is called finite-dimensional. Remark 2.32.1. In the remaining part of section we shall call an associative algebra just algebra and assume it is finite-dimensional. Definition 2.33. An division algebra is an algebra in which the underlying ring is a division ring.

Example 2.33.1. R is a one-dimensional vector space over R and a ﬁeld, hence an associative division algebra over R of dimension 1. Example 2.33.2. The ﬁeld C and the vector space C = R ⊕ iR over R form an associative division algebra over R of dimension 2. We have seen the quaternion ring in Proposition 2.5 and the quaternion vector space in Corollary 2.19, combining both results yields the following proposition.

Proposition 2.34. The ring H from Proposition 2.5 is a real associative division algebra. This algebra is called the quaternion algebra.

Proof. Corollary 2.19 yields that the ring H has a real vector space structure. Axioms (A1) and (A2) follows then from the ring axioms. For (A3), let λ ∈ R, then λ is in the centre of H so it commutes with all elements. Thus for all u, v ∈ H, λ(uv) = (λu)v = (uλ)v = u(λv) where the first and third equality follows from the associativity of H. Remark 2.34.1. Note that the dimension of the quaternion algebra is 4. Definition 2.35. Let A be an algebra over a field F . The minimal polynomial of a ∈ A is a monic polynomial ma ∈ F [X] such that ma(a) = 0 and for every other monic non-zero polynomial P ∈ F [X] such that P (a) = 0 it holds that deg ma ≤ deg P . Proposition 2.36. Every element a in an division algebra has a unique minimal polynomial ma. Moreover, ma is irreducible. Proof. For existence, let n be the dimension of the algebra, take a ∈ A and consider the set {ai : i = 0, . . . , n} ⊆ A, then this set consists of n + 1 vectors so they must be linearly dependent and hence there exists λ0, . . . λn ∈ F not 0 n all zero, such that λ0a + ... + λna = 0. Let m be the larges index such that 1 m λm 6= 0 then the polynomial P (X) = (λmX + ... + λ0) is monic and has a λm as a root. Since the degree of a polynomial is bounded from below there must exist a polynomial with smallest degree with this property.

9 For uniqueness, let A be an algebra over the ﬁeld F and assume that 0 ma, ma ∈ F [X] are minimal polynomials of a ∈ A and denote theirs degree n Pn−1 i 0 n Pn−1 i by n, then ma = X + i=0 λiX and ma = X + i=0 µiX for some λi:s 0 and µi:s is F . Now the polynomial (ma − ma) has degree at most n − 1 and 0 0 0 (ma − ma)(a) = ma(a) − ma(a) = 0 which implies that ma − ma is the zero 0 polynomial hence ma = ma and thus the minimal polynomial is unique. For irreducibility, if ma(X) = P (X)Q(X) for some P,Q ∈ F [X] then 0 = ma(a) = P (a)Q(a) so either P (a) = 0 or Q(a) = 0 meaning that at least one of P and Q must have degree greater than or equal to ma and hence the other must be constant.

Proposition 2.37. Let A be an division algebra over R then the minimal polynomial of a ∈ A is ( X − a if a ∈ R ma(X) = . X2 − 2pX + q for some p, q ∈ R with p2 < q if a∈ / R

Proof. The case when a ∈ R is clear and if a∈ / R then for every polynomial Pr of degree 1, Pr(X) = X − r for some r ∈ R and Pr(a) = 0 implies a = r so the minimal polynomial can not have degree one and since the minimal polynomial is irreducible Corollary 2.29 yields the other case.

3 Frobenius Theorem

We have now the necessary background to prove Frobenius theorem. But ﬁrst we shall prove a lemma that will simplify the proof. The elements of a real division algebra can be characterised by their minimal polynomial. Proposition 2.37 has shown that the minimal polynomial of an element a is, X − a if a is real and X2 − 2pX + q with p2 < q otherwise which yields the following lemma. Lemma 3.1. Let A be a ﬁnite-dimensional real associative division algebra then each a ∈ A can be written on the form a = x + y where x ∈ R and y = 0 or 2 y ∈ R<0. Proof. Let a ∈ A, if a is real then the condition hold trivially so assume a∈ / R 2 then by Proposition 2.37 the minimal polynomial of a is ma(X) = X −2pX +q with p2 − q < 0, moreover let b = a − p∈ / R then

2 2 2 0 = ma(a) = ma(b + p) = (b + p) − 2p(b + p) + q = b − p + q which implies that b2 = p2 − q < 0 and a = p + b with p ∈ R and b2 < 0. Theorem 3.2 (Frobenius theorem). The only ﬁnite-dimensional real associative division algebras, up to isomorphism, are (i) R, (ii) C and (iii) H.

10 Proof. Let A be a ﬁnite-dimensional real associative division algebra and consider the subset A0 consisting of all elements whose square is real and non- positive. Note that a∈ / A0 for all non-zero a ∈ R. We shall now prove that A0 is a subspace of A. It is closed under scalar multiplication since if v ∈ A0 and λ ∈ R then λ2 ≥ 0 and (λv)2 = λ2v2 ≤ 0 so λv ∈ A0. In order to see that A0 is closed under addition we take non-zero u, v ∈ A0 and show that u + v ∈ A0. If u = λv for some λ ∈ R then u + v = (λ + 1)u ∈ A0. Otherwise, assume that u and v are linearly independent. Note that u, v, 1 are now pair-wise linearly independent. In fact we claim that all three are linearly independent. Consider the equation νu + µv + λ1 = 0 with ν, µ, λ ∈ R. Then −νu = µv + λ and ν2u2 = (µv + λ)2 = (µv)2 + 2λµv + λ2.

The left hand side is real and so is (µv)2 + λ2 but λµv is in A0 meaning that λµv is real only if λµ = 0 so either λ = 0 or µ = 0. But then since u, v, 1 are pair-wise linearly independent λ = µ = ν = 0 is the only solution, thus 1, u, v are linearly independent. It follows that u + v, u − v ∈ A \ R. By Proposition 2.37 the minimal 2 polynomials of both u + v and u − v have degree 2. Let m1(X) = X + p1X + q1 2 and m2(X) = X + p2X + q2 be the minimal polynomials of u + v and u − v, respectively. Then

2 2 ) 0 = m1(u + v) = u + uv + vu + v + p1u + p1v + q1 2 2 ⇒ 0 = m2(u − v) = u − uv − vu + v + p2u − p2v + q2 2 2 0 = (p1 + p2)u + (p1 − p2)v + (2u + 2v + q1 + q2)1.

Now the linear independence of u, v, 1 yields that

2 2 (p1 + p2) = (p1 − p2) = (2u + 2v + q1 + q2) = 0

2 and thus p1 = p2 = 0. Hence m1(X) = X + q1, with q1 > 0 by Proposition 2 0 2.37. It follows that (u + v) = −q1 < 0 and thus u + v ∈ A which completes the proof of the claim that A0 is a subspace of A. Lemma 3.1 showed that every u ∈ A can be written on the form u = a + b with a ∈ R and b ∈ A0 hence A = R ⊕ A0. Let Q : A0 → R be deﬁned by Q(v) = −v2 then Q(0) = 0 and Q(λv) = −λ2v2 > 0 for all non-zero scalars λ and non-zero v in A0. Moreover B(u, v) := Q(u+v)−Q(u)−Q(v) = −(uv +vu) is symmetric and

B((λu + µv), w) = −(λu + µv)w − −w(λu + µv) = −λ(uw + wu) − µ(vw + wv) = λB(u, w) + µB(v, w), shows that B is bilinear, thus Q is a quadratic form and B is positive-deﬁnite. In order to complete the proof we shall now show that the only possible dimensions of A0 are 0, 1 and 3. If A0 = {0} then A = R ⊕ {0} = R which corresponds to the case (i). Now assume A0 is non-trivial. Then there exists an i ∈ A0 with 1 = Q(i) = −i2 and iR ⊆ A0 is a subspace. If A0 = iR then

11 A = R ⊕ iR = C, which corresponds to the case (ii). Finally, if iR 6= A0 then there exists a vector j, orthogonal to i with respect to B such that j2 = −1. Taking the bilinear form of i and j gives 0 = B(i, j) = −(ij + ji) which implies that ij = −ji hence there exists a k = ij ∈ A and k2 = ijij = −jiij = j2 = −1 thus k ∈ A0 but

−B(i, k) = ik + ki = iij − jii = −j + j = 0 and −B(j, k) = jk + kj = jji − ijj = −i + i = 0, meaning that k is orthogonal to both i and j and thus A0 can not have dimension 2. It is left to show that A0 must now be equal to span{i, j, k}. Let v ∈ A0 and consider B(v, i) B(v, j) B(v, k) ` = v − i − j − k. B(i, i) B(j, j) B(k, k) Then ` is orthogonal to i, j and k, since

B(v, i) B(`, i) = B(v, i) − B(i, i) = 0 B(i, i) and similarly for j and k. Moreover, the orthogonality yields i` = −`i, j` = −`j, k` = −`k and so

k` = −`k = −`ij = i`j = −ij` = −k`, which implies that 0 = ` so

B(v, i) B(v, j) B(v, k) v = i + j + k ∈ span{i, j, k} . B(i, i) B(j, j) B(k, k)

Thus A0 = span{i, j, k}. Moreover, it holds that i2 = j2 = k2 = −1 = ijk hence the last case is given by A = R ⊕ iR ⊕ jR ⊕ kR = H by Remark 2.6.1.

4 Classiﬁcation of ﬁnite-dimensional real division algebras

Before a classification can be given the definition of an arbitrary division algebra is in order. Definition 4.1. An algebra A is called a division algebra if for all non-zero a ∈ A the maps La : A×A → A defined by v 7→ av and Ra : A×A → A defined by v 7→ va are invertible. This generalises the notion of an associative division algebra. Frobenius theorem yields that the only possible dimensions of a finite-dimensional real associative algebras are 1, 2 or 4, by weakening the assumption of associativity we get a new type of algebras.

12 Deﬁnition 4.2. An alternative algebra is an algebra A such that

(uu)v − u(uv) = 0 = (uv)v − u(vv) holds for all u, v ∈ A.

By Zorn’s theorem [12] requiring that the algebra is alternative instead of associative yields only one additional algebra, the octonions which are 8- dimensional. A natural question to ask is which dimension an finite-dimensional real division algebra can have? The answer turn out to by 1, 2, 4 or 8 and has been proved by Bott and Milnor in [4, Corollary 1]. The classification of all finite-dimensional real division algebras is still an open problem. The 1-dimensional case has only the trivial solution R [6]. In this section a solution of the 2-dimensional case, when the number of idempotent elements in not three, will be given.

4.1 2-dimensional division algebras Another approach to classification of real division algebras is rather than fixing the properties of the algebra instead fix it’s dimension. In this subsection a almost complete classification of 2-dimensional real division algebras will be given based on the work of Althoen and Kugler [1]. In this subsection every algebra will be assumed to be a 2-dimensional real algebra if not stated otherwise. An 2-dimensional real algebra if fully determined by the multiplication of the basis, since then the properties (A1), (A2) and (A3) from Definition 2.30 yields the multiplication of arbitrary elements. Thus such algebras can be described by a multiplication table as in Table 1 where {u, v} is some basis and λij, µij ∈ R for i, j ∈ {1, 2}.

· u v u λ11u + µ11v λ12u + µ12v v λ21u + µ21v λ22u + µ22v

Table 1

Remark 4.2.1. Although a basis is defined as a set of vectors in this subsection the order will of the essence and the basis {u, v} and {v, u} will be considered to be different. Definition 4.3. Two multiplication tables as in Table 2 are called identical if 0 0 λij = λij and µij = µij for all i, j ∈ {1, 2}. Lemma 4.4. Let A and A0 be two algebras. If there exist bases such that the multiplication tables of A and A0 are identical then A and A0 are isomorphic. Conversely, if ϕ : A → A0 is an isomorphism and {u, v} a basis for A then the multiplication tables with basis {u, v} of A and {ϕ(u), ϕ(v)} of A0 are identical.

13 · u v · u0 v0 0 0 0 0 0 0 0 0 0 u λ11u + µ11v λ12u + µ12v u λ11u + µ11v λ12u + µ12v 0 0 0 0 0 0 0 0 0 v λ21u + µ21v λ22u + µ22v v λ21u + µ21v λ22u + µ22v (a) (b)

Table 2

Proof. Let A be an algebra given by Table 2a and A0 be an algebra given by Table 2b. Assume the tables are identical and let ϕ : A → A0 be deﬁned by ϕ(λu+µv) = λu0 +µv0 for all λ, µ ∈ R then clearly ϕ(λu+µv) = λϕ(u)+µϕ(v),

0 0 0 0 ϕ(uv) = ϕ(λ12u + µ12v) = λ12u + µ12v = u v = ϕ(u)ϕ(v) and the other cases can be calculated in a similar way. Thus ϕ is an isomorphism. Conversely, if ϕ : A → A0 is an isomorphism then

2 2 ϕ(u) = ϕ(u ) = ϕ(λ11u + µ11v) = λ11ϕ(u) + µ11ϕ(v). and the other cases follows in a similar way. The classification of 2-dimensional real division algebras is based on the number of idempotent element hence the following definition is in order. Definition 4.5. A non-zero element u of an algebra is called idempotent if u2 = u. Lemma 4.6. An isomorphism maps idempotent elements to idempotent elements. Proof. Let u be idempotent and ϕ an isomorphism then ϕ(u) = ϕ(u2) = ϕ(u)2.

Proposition 4.7. A 2-dimensional real division algebra has at least one idempotent element. For proof see [1, Theorem 2]. Thus by choosing an appropriate basis the multiplication table of every algebra can be written on the form of Table 3.

· u v u u λ12u + µ12v v λ21u + µ21v λ22u + µ22v

Table 3

Deﬁnition 4.8. Let A(λ12, µ12, λ21, µ21, λ22, µ22) denote an algebra with multiplication table on the form of Table 3.

14 Proposition 4.9. An algebra given by Table 3 is a division algebra if and only

2 λ12 µ12 λ21 µ21 if (µ22 − A1) < 4µ12A2 where A1 = and A2 = . λ21 µ21 λ22 µ22 This lemma can be proved by applying Cramer’s Rule, for details see [1, Theorem 3]. Another useful fact in the following proposition [1, Proposition on p. 629].

Proposition 4.10. An algebra has exactly one idempotent element if and only 2 if (λ12 + λ21 − µ22) < 4λ22(1 − µ12 − µ21). Assume there is an algebra that has at least two idempotent elements u and v. Then the elements must be linearly independent since v = λu implies that v = v2 = λ2u2 = λ2u = λv so λ = 1 and u = v. Thus {u, v} is a basis and the multiplication table is

· u v u u λ12u + µ12v . v λ21u + µ21v v

Table 4

Lemma 4.11. In a 2-dimensional real division algebra given by the Table 4, (λ12 + λ21)(µ12 + µ21) 6= 1. Proof. An algebra given by Table 4 must satisfy

2 2 1 + (λ12µ21) + (λ21µ12) − 2(λ12µ21) − 2(λ21µ12) − 2(λ12µ21)(λ21µ12) < 0, (2) by Proposition 4.9. Hence λ12, λ21, µ12, µ21 are all non-zero. Let A = λ12µ21 2 and B = λ21µ12. Then (2) simpliﬁes to (A + B − 1) − 4AB < 0. If B A (λ12 + λ21)(µ12 + µ21) = 1, then substitution µ12 = and µ21 = yields λ21 λ12

λ λ λ 2 λ B 12 + A + B + A 21 = 1 ⇔ B 12 + (A + B − 1) 12 + A = 0 λ21 λ12 λ21 λ21 so λ12 is a root of a quadratic equation. Thus the discriminant λ21 (A + B − 1)2 − 4AB is non-negative which contradicts (2). The ﬁrst step in classiﬁcation of 2-dimensional algebras is the following theorem. Theorem 4.12. A 2-dimensional real division algebra has exactly one, two or three idempotent elements. Proof. By Proposition 4.7 all such algebras have at least one idempotent element, hence let A be such algebra with basis {u, v} and at least two idempotent

15 elements. Then the multiplication in A can be written on the form of Table 4. An arbitrary element xu + yv in A is idempotent if

2 2 2 xu + yv = (xu + yv) = x u + (λ12 + λ21)xyu + (µ12 + µ21)xyv + y holds, which yields the system

( 2 x = x + (λ12 + λ21)yx 2 (3) y = y + (µ12 + µ21)xy ( ( 0 0 and the solutions are x = and y = . 1 − (λ12 + λ21)y 1 − (µ12 + µ21)x If x = 0 or y = 0 then the only possible solutions are (x, y) = (0, 0), (0, 1) and 1−Λ (1, 0). Otherwise, let Λ = λ12 + λ21 and M = µ12 + µ21 then x = 1−ΛM and 1−M y = 1−ΛM is the only possible solution by Lemma 4.11. Thus the idempotent 1−Λ 1−M elements are u, v and 1−ΛM u + 1−ΛM v. Note that they are not necessarily 1−Λ 1−M distinct and 1−ΛM u + 1−ΛM v is non-zero by Lemma 4.11. By Lemma 4.6 two algebras can only be isomorphic if they have the same number of idempotent elements, which together with Theorem 4.12 yields three classes of algebras, 1-, 2- and 3-idempotent algebras. Let A be an algebra with at least two idempotent elements u, v. Then the multiplication can be written on the form Table 4 with basis {u, v}. From proof of Theorem 4.12 we know that A to has exactly two idempotent elements if and 1−Λ 1−M only if 1−ΛM u + 1−ΛM v is equal to u or v. Thus exactly one of Λ and M must be equal to 1 in order for A to have exactly two idempotent elements. If A is a 2-idempotent algebra with M = 1 then a change of basis to {v, u} yields a multiplication table of the form of Table 4 with Λ = 1. Thus a multiplication table with Λ = 1 can be considered to be the canonical multiplication table for a 2-idempotent algebra. The following theorem classiﬁes all 2-idempotent algebras. Theorem 4.13. Two 2-idempotent 2-dimensional real division algebras are isomorphic if and only if theirs multiplication tables are identical when written on the form of Table 4 with λ12 + λ21 = 1. Proof. Let A and A0 be such algebras. By the discussion above every 2-idempotent algebra can be written on the form of Table 4 with Λ = 1, let {u, v} and {u0, v0} be the basis of A and A0 in which the multiplication is on the desired form. If the algebras are isomorphic then there exists an isomorphism ϕ which must map u to either u0 or v0. In the ﬁrst case v is then mapped to v0 and Lemma 4.4 yields that the tables with basis {u, v} and {u0, v0} are identical. If ϕ(u) = v0 then by the same lemma the tables {u, v} and {v0, u0} are identical, 0 0 0 0 0 0 but then λ12 = µ21, λ21 = µ12 so µ12 + µ21 = 1 but by the assumption λ12 + λ21 is also 1 which contradicts Lemma 4.11 so ϕ is not an isomorphism in this case. Conversely, if the tables are identical then by Lemma 4.4, A and A0 are isomorphic.

16 Corollary 4.14. The set

2 D2 = {A(a, c, 1 − a, d, 0, 1) : c + d 6= 1, (1 − ad + (1 − a)c) < 4c(1 − a)} is a complete and irredundant list of 2-idempotent 2-dimensional real division algebras. The case of three idempotent elements is similar and hence shall be considered next. Let A be a 3-idempotent algebra with idempotent elements u, v and basis {u, v}. Then as before the multiplication can be written on the form of Table 4 and ΛM 6= 1. As seen in the proof of Theorem 4.12 the third idem- 1−Λ 1−M potent element is then w = 1−ΛM u+ 1−ΛM v, meaning that neither Λ nor M can be equal to one. But now {u, w} is also a basis with multiplication of the form of Table 4. In fact a 3-idempotent algebra has 6 bases with such multiplication. A change of basis to {u, w} yields that (1 − Λ)u + (1 − M)(λ u + µ v) uw = 12 12 = ν u + µ w, 1 − ΛM 12 12

(1−Λ)+(1−M)λ12−(1−Λ)µ12 λ12µ21−(µ12−1)(λ21−1) where ν12 = 1−ΛM = ΛM−1 and similarly ν21 is ν12 with all indices changed from 12 to 21 and vice versa. Which yield the three multiplication tables in Table 5, the other three are given by simply reversing the order of the basis vectors in these tables.

· u v · u w u u λ12u + µ12v u u ν12u + µ12w v λ21u + µ21v v w ν21u + µ21w w (a) (b)

· v w v v ν12u + λ21w w ν21u + λ12w w (c)

Table 5

Proposition 4.15. If A(λ12, µ12, λ21, µ21, 0, 1) is a 3-idempotent division algebra then it is isomorphic to A(λ21, µ21, λ12, µ12, 0, 1), A(ν12, µ12, ν21, µ21, 0, 1), A(ν21, µ21, ν12, µ12, 0, 1), A(ν12, λ21, ν21, λ12, 0, 1) and A(ν21, λ12, ν12, λ21, 0, 1) λ12µ21−(µ12−1)(λ21−1) where ν12 = ΛM−1 and ν21 is ν12 with all indices changed from 12 to 21 and vice versa.

Proof. If A(λ12, µ12, λ21, µ21, 0, 1) has basis {u, v} and the third idempotent element is w then a change of basis to {v, u}, {u, w}, {w, u}, {v, w} and {w, v}, respectively together with Lemma 4.4 yields the proposition. A complete list of 3-idempotent algebras is given by the following theorem.

17 Theorem 4.16. Two 3-idempotent 2-dimensional real division algebras are isomorphic if and only there is a correspondence between the idempotent elements and the corresponding multiplication tables are identical. Proof. Let A and A0 be such algebras the ﬁrst one with idempotent elements u, v and w. If they are isomorphic and ϕ is an isomorphism then ϕ(u), ϕ(v) and ϕ(w) are the idempotent elements in A0. Now, the tables with basis {u, v} and {ϕ(u), ϕ(v)} are identical, {u, w} and {ϕ(u), ϕ(w)} are identical and also {v, w} and {ϕ(v), ϕ(w)} are identical by Lemma 4.4. Conversely, if the tables are identical then by Lemma 4.4, A and A0 are isomorphic. Corollary 4.17. The set

2 D3 = {A(a, c, b, d, 0, 1) : a+b 6= 1 6= c+d, (a+b)(c+d) 6= 1, (1−ad+bc) < 4cb} is a complete list of 3-idempotent 2-dimensional real division algebras.

Note that an algebra in D3 can be isomorphic to at most 5 other algebras in D3. The last case to consider is when the algebra has exactly one idempotent element, call it u. As before there is a multiplication table on the form of Table v 3 but the 1-idempotent case requires λ22 6= 0 since otherwise would be µ22 idempotent. Proposition 4.18. Given a 2-dimensional real division algebra A with one idempotent element u and multiplication given by Table 3 if µ12 + µ21 6= 0 then there exists a unique up to sign w such that {u, w} is a basis in which w2 = ±u. It can be proven by letting w = xu + yv with y 6= 0 and solving the equation (xu + yv)2 = ±u for details see [1, Theorem 7].

Lemma 4.19. In an algebra given by Table 3 the coeﬃcients µ12 and µ21 are invariant under the change of basis to {u, w} for all w linearly independent to u. Proof. Consider such algebra with basis {u, v} and let w = xu + yv. In order for w to be linearly independent to u, y must be non-zero. Then

uw = u(xu + yv) = xu + yλ12u + yµ12v = (x + yλ12 − xµ12)u + µ12w,

wu = (xu + yv)u = xu + yλ21u + yµ21v = (x + yλ21 − xµ21)u + µ21w.

This lemma makes the following deﬁnition well-deﬁned.

Deﬁnition 4.20. An algebra given by Table 3 is quasi-complex if µ12 +µ21 6= 0. Hence a quasi-complex algebra can be written on the form of Table 6. The following theorem classiﬁes all quasi-complex algebras.

18 · u v u u λ12u + µ12v v λ21u + µ21v ±u

Table 6

Theorem 4.21. Two 1-idempotent quasi-complex 2-dimensional real division algebras given by Table 6 with basis {u, v} and {u0, v0} are isomorphic if and only if the multiplication table are identical or the multiplication tables with basis {u, v} and {u0, −v0} are identical. Proof. Let A and A0 be such algebras where A has basis {u, v} and let ϕ be an isomorphism then ϕ(u) = u0 and ϕ(v)2 = ϕ(v2) = ϕ(±u) = ±u0 so ϕ(v) = ±v0 by Proposition 4.18 and so {u0, ϕ(v)} is equal to {u0, v0} or {u0, −v0} and then by Lemma 4.4 the multiplication tables with basis {u, v} and {u, ϕ(v)} are identical. Conversely, note that the algebra given by the multiplication table with basis {u0, −v0} is just a change of basis of the algebra with basis {u0, v0}. Thus by Lemma 4.4, A and A0 are isomorphic. Corollary 4.22. The set

2 D1a = {A(a, c, b, d, e, 0) : c + d 6= 0, e = ±1, (bc − ad) < −4cde, (a + c)2 < 4e(1 − c − d)} is a complete and irredundant list of 1-idempotent quasi-complex 2-dimensional real division algebras. Another type of 1-idempotent algebras are algebras with a basis such that λ12 = λ21 = 0. In order to ﬁnd such basis consider w = xu + yv with y 6= 0.

uw = xu + y(λ12u + µ12v) = ((1 − µ12)x + yλ12)u + µ12w and similarly for wu which yields the following system of equations, ( (1 − µ )x + yλ = 0 12 12 (4) (1 − µ21)x + yλ21 = 0

This system has always the solution x = y = 0. Moreover, the solution is unique

(1 − µ12) λ12 whenever = (λ12µ21 − λ21µ12) − λ12 + λ21 6= 0. A non-trivial (1 − µ21) λ21 solution exists if and only if (λ12µ21 − λ21µ12) = λ12 − λ21. For a commutative algebra the condition is always true which explains the following deﬁnition. Deﬁnition 4.23. A 2-dimensional real division algebra is called quasi-commutative if λ12µ21 − λ21µ12 = λ12 − λ21. Proposition 4.24. If a 1-idempotent 2-dimensional real division algebra is quasi-commutative but not quasi-complex then the algebra has basis {u, v} in which the multiplication is as in Table 7.

19 · u v u u µ12v v −µ12v u + µ22v

Table 7

Proof. Let u be the idempotent element and the multiplication be given by Table 3. Since the algebra is non-quasi-complex, µ12 + µ21 = 0 and the condition for quasi-commutativeness simpliﬁes to µ12(λ12 +λ21) = λ21 −λ12. If λ12 +λ21 = 0 then λ12 = λ21 = 0. Letting w = v yields multiplication as in Table 8.

· u w u u µ12w w −µ12w λ22u + µ22w

Table 8

Otherwise, adding both equations in the system (4) yields 2x+(λ12 +λ21)y = 0. Then x = 1 and y = −2 is a solution. Let w = xu + yv. The change of λ12+λ21 basis to {u, w} yields multiplication on the form of Table 8. Moreover, in a division algebra with multiplication as in Table 8, 2 (−µ22) < 4λ22 must√ hold by Proposition 4.10. Hence λ22 is positive. The change of basis v = w/ λ22 yields that w2 µ w µ v2 = = u + 22 = u + √ 22 v, λ22 λ22 λ22 while preserving the other multiplications. Theorem 4.25. Two 1-idempotent non-quasi-complex quasi-commutative 2- dimensional real division algebras are isomorphic if and only if when the multiplication table are written on the form of Table 7 the tables are identical. Proof. The theorem follows from Proposition 4.24 and Lemma 4.4. Corollary 4.26. The set

2 2 2 D1b = A(0, c, 0, −c, 1, f): f < 4, f < 2c is a complete and irredundant list of 1-idempotent non-quasi-complex quasi-commutative 2-dimensional real division algebras. The last case is algebras that are neither quasi-complex nor quasi-commutative and they are classiﬁed by the following theorem. Theorem 4.27. A 1-idempotent non-quasi-complex non-quasi-commutative 2- dimensional division algebra has a basis in which the multiplication table is of the form of Table 9. Moreover two such algebras are isomorphic if and only if theirs multiplication tables are identical.

20 · u v u u u + µ12v v u − µ12v λ22u + µ22v

Table 9

Proof. Consider such algebra with multiplication as in Table 3 then since it is not quasi-complex µ12 = −µ21. The change of basis is given by the system ( (1 − µ )x + yλ = 1 12 12 (5) (1 + µ12)x + yλ21 = 1 which has a unique solution since the non-quasi-commutativeness yields that

(1 − µ12) λ12 6= 0. In order for the solution to be a proper change of basis (1 + µ12) λ21 1 1 y must be non-zero. If y = 0 then = x = , hence µ12 must be 0. 1+µ12 1−µ12 Conversely, if µ12 = 0 then the system of equations yields y(λ12 − λ21) = 0 and non-quasi-commutativeness yields y = 0. So y = 0 if and only if µ12 = 0. But if µ12 = 0 then u(λ12u − v) = λ12u − λ12u = 0 so λ12u = v and hence {u, v} is not a basis. Thus there always exists a change of basis in which the multiplication is as in Table 9. Moreover, the uniqueness implies that two such algebras are isomorphic if and only if the multiplication tables are identical.

Corollary 4.28. The set

2 2 D1c = {A(1, c, 1, −c, e, f): c 6= 0 6= e, f < 4c(f + ce), (2 − f) < 4e} is a complete and irredundant list of 1-idempotent non-quasi-complex quasi-commutative 2-dimensional real division algebras.

Thus the set D = D1a ∪ D1b ∪ D1c ∪ D2 ∪ D3 is a complete list of all 2- dimensional real division algebras where the only redundant algebras have exactly three idempotent elements. A complete and irredundant classiﬁcation is given in [10] although it uses a more general theory of division algebras not discussed in this paper.

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